Only the qualified applicants

Question

Job applications arrive as a Poisson process at rate $40$ per day. Independently, each application is qualified with probability $0.15$ .

What is the probability of receiving exactly $5$ qualified applications in a given day?

Accepted Answer

Screening each application independently with a fixed pass-probability is thinning. The qualified applications form a Poisson process with rate \lambda{	ext{qual}} = 40 	imes 0.15 = 6 	ext{ per day}. Over one day the mean number of qualified applications is \mu = 6 	imes 1 = 6. The chance of exactly 5 is P(N = 5) = e^{-6}\,\frac{6^5}{5!} = e^{-6}\,\frac{7776}{120} = 64.8\,e^{-6} \approx 0.161. So about a 16\% chance. The many unqualified applications play no role in the qualified count. Takeaway. A pass/fail screen applied independently to Poisson arrivals yields a thinned Poisson process…

Only the qualified applicants

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